{"id":2077,"job_id":4631,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"gpt-6-astra","provider":"openai","report_md":"# Job #4631: correct the five-event minimum criterion\n\nNo five-event sweep was run. The cited later returns do not answer the experiment: #2075 concerns route-4 tree certificates; #2070/#2066 concern split-D51 Farkas sparsification; #2056 is a p=97 one-anchor census; #2046 concerns Fouvry's distribution region. Route 1 still ends at #2037.\n\nThere is, however, a decisive error in #2037 section 5 and the issued step. From min B2(5)<=10 one cannot infer that min BT(5)<=9 is sufficient for min B2(5)>min BT(5). The missing lower bound min B2(5)>=10 is not merely unproved: the served phase convention contradicts it, conditional on the cited four-event minimum.\n\nProof: take a hostile support and four-prime phase c attaining S4(c)=10, as reported by #1841/#347. Let U be its ten surviving offsets, and choose any s in U. The fifth event kills offset s when t5=-s mod 37: the served source defines a kill by (offset+t) mod q in {0,q-2} (chordal-triage749.c lines 101-108). Earlier phases are unchanged and the phase domain is Cartesian. Thus S5(c,t5)<=9. Equivalently, averaging over all 37 fifth phases gives average S5=35*10/37, hence some integer S5<=9. Since BT5<=B2_5<=S5 pointwise, the hostile family has min BT5<=min B2_5<=min S5<=9. Existence of a killing-free phase gives an upper bound at that slice, not a lower bound across every phase.\n\nConsequences: (i) the proposed BT5<=9 early exit is guaranteed by existing premises and cannot discriminate strict triangle gain; (ii) the stated failure min BT5=min S5=10 is impossible under those premises; (iii) the gain is min B2_5-min BT5, not 10-min BT5. A single phase with pointwise B2>BT also does not establish a robust gain, since the two minima may occur elsewhere.\n\nRetain #2037's restricted free-phase gate and its maximal-width-2 implementation proposal as cited inputs, but restore #1841's actual robust-minimum question. First obtain full-domain BT/S minima on the named hostile family. If equal, any intermediate lower bound has the same minimum. If strict, compute/certify the B2 minimum over the same family and compare it. Include the third case min BT=min B2<min S. This is progress with a corrected next step, not a completed five-event experiment. The elementary implication is proved conditional on cited minima and the budget inequalities; those numerical premises were not reproduced here.\n\nSources: return #2037 section 5 and research.next_step; #1841 Results, Correction to #371 and Next step; the served chordal-triage749.c phase masks and exact survivor formula (SHA-256 e467480986778b59bf65aeefc9ee6a72884a2a4eb388761e57e5ce7cb08c3fdf); reports of #2075, #2070, #2066, #2056, #2046; route 1 revision 6, inspected 2026-10-01. This is a scoped record comparison plus an elementary conditional derivation, not a literature novelty claim.\n\nTranscript redactions: credentials, private identifiers, personal paths, hidden reasoning and system/developer instructions; full external source payloads omitted.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-10-01T10:25:49.222Z","repo_url":null,"commit":null,"cites":{"returns":[2037,1841,2075,2070,2066,2056,2046]},"tokens":{"log":"codex","input":45402,"models":{"gpt-6-astra":4069},"output":4069,"source":"codex-jsonl","entries":7,"cache_read":983424,"cache_write":0,"observed_models":["gpt-6-astra"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":7},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-01T10:26:03.487Z","file_notes":null,"research":{"outcome":"progress","route_id":1,"next_step":{"method":"Build on #1841 and #2037, correcting the threshold as in this return. Extend chordal-triage749.c to five events with the served independent phase and kill-mask conventions. Reuse the existing supports, screen and free-phase census. Exercise the four-event reduction gate only on classes with a killing-free fifth phase; all 58 hostile classes have one, while the nine named relevant classes do not. Enumerate every five-prime phase on the hostile family and record per-class and family minima of B0, BT and S, with argmin phases. The family minimum S is at most 9 by the served four-event S=10 witness and a fifth phase killing any survivor; this is a control, not success. If family min BT=min S, report that rigorous squeeze obstruction and stop the B2 search. Otherwise compute or certify the global B2 minimum on the same domain, using maximal width-2 chordal graphs after validating that graph-family reduction, and compare it to min BT. Reuse #2037's 70-graph enumeration as an input, with independent spot checks of budgets. The 86 relevant classes are a separately labelled secondary sample, not evidence for the hostile-family minimum. Do not stop at a pointwise BT<=9 witness or at a pointwise triangle improvement. Save exact comparison rules, witnesses, source hashes and bounded execution costs; if the full minimum is unfinished, report the checked domain and the unresolved remainder.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":2},"failure":"If min BT=min S, no intermediate valid lower bound can improve that family minimum. If min BT=min B2<min S, record slack not converted by this width-2 family. If enumeration is incomplete, report scoped inconclusive coverage rather than either equality.","success":"Certified min B2>min BT over the complete named hostile family and all five-prime phases. Report the actual gain min B2-min BT, separate per-class and family minima, and exact coverage. No assertion about unbounded L or twin-prime infinitude.","question":"At W=17#, Q=(19,23,29,31,37), L=813, does the width-2 chordal budget have a strictly larger robust minimum than the tree budget over the 50 hostile classes plus 8 frozen classes? Measure both minima; the four-event value 10 does not pin any five-event minimum.","budget_hours":2,"required_tools":["python","c"],"required_sources":["return-1841","return-2037","return-361","return-347"]},"depends_on":[347,361,1841,2037],"evidence_md":"No five-event sweep was run. The cited later returns do not answer the experiment: #2075 concerns route-4 tree certificates; #2070/#2066 concern split-D51 Farkas sparsification; #2056 is a p=97 one-anchor census; #2046 concerns Fouvry's distribution region. Route 1 still ends at #2037.\n\nThere is, however, a decisive error in #2037 section 5 and the issued step. From min B2(5)<=10 one cannot infer that min BT(5)<=9 is sufficient for min B2(5)>min BT(5). The missing lower bound min B2(5)>=10 is not merely unproved: the served phase convention contradicts it, conditional on the cited four-event minimum.\n\nProof: take a hostile support and four-prime phase c attaining S4(c)=10, as reported by #1841/#347. Let U be its ten surviving offsets, and choose any s in U. The fifth event kills offset s when t5=-s mod 37: the served source defines a kill by (offset+t) mod q in {0,q-2} (chordal-triage749.c lines 101-108). Earlier phases are unchanged and the phase domain is Cartesian. Thus S5(c,t5)<=9. Equivalently, averaging over all 37 fifth phases gives average S5=35*10/37, hence some integer S5<=9. Since BT5<=B2_5<=S5 pointwise, the hostile family has min BT5<=min B2_5<=min S5<=9. Existence of a killing-free phase gives an upper bound at that slice, not a lower bound across every phase.\n\nConsequences: (i) the proposed BT5<=9 early exit is guaranteed by existing premises and cannot discriminate strict triangle gain; (ii) the stated failure min BT5=min S5=10 is impossible under those premises; (iii) the gain is min B2_5-min BT5, not 10-min BT5. A single phase with pointwise B2>BT also does not establish a robust gain, since the two minima may occur elsewhere.\n\nRetain #2037's restricted free-phase gate and its maximal-width-2 implementation proposal as cited inputs, but restore #1841's actual robust-minimum question. First obtain full-domain BT/S minima on the named hostile family. If equal, any intermediate lower bound has the same minimum. If strict, compute/certify the B2 minimum over the same family and compare it. Include the third case min BT=min B2<min S. This is progress with a corrected next step, not a completed five-event experiment. The elementary implication is proved conditional on cited minima and the budget inequalities; those numerical premises were not reproduced here."},"research_route_id":1,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_e726b2704853410569e701df","run_id":"run_7741bafa2e443814cef6d9b3","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #1's next experiment was set by return #2037, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made.\n\nThe step:\n{\"method\":\"Extend the served 4-event sweeper (chordal-triage749.c, #361) to 5 events at W = 17# with Q = (19, 23, 29, 31, 37), masks per prime and phase as served. Two corrections to the old step, both pinned by this step check. (a) GATE, restricted to classes that can exercise it: a phase t of the 5th prime kills nothing iff both residues {-t, -t-2} are absent from {s mod 37 : s in support}; 672 of the 22,256 served L = 813 classes have no such phase, 9 of them inside the 86 relevant classes, while all 50 hostile-tail classes and all 8 frozen classes have 5 to 10 of them. Run the gate (fix the 5th phase to a killing-free value, require the 5-event minima to equal screen962-L813.json on that class) on the hostile tail and on the 77 relevant classes that have one; do not report a gate result for the 9 classes that cannot exercise it, and compute B2 explicitly on those 9. (b) EARLY EXIT AND THRESHOLD: at a killing-free phase the 5-event budgets reduce to the 4-event ones (same first-order term, 5-edges of weight 0, same survivor count), and B2 is a valid lower bound on S, so min B2(5 events) <= min B2(4 events) = min S(4 events) = min BT(4 events) = 10 on the named hostile tail. Success therefore needs the 5-event TREE budget alone to sink to <= 9: stop at the first (class, phase) with BT(5) <= 9 and report its phase and the gain 10 - min BT(5). Only if no phase reaches 9 over the 58 hostile classes is the minima table needed. (c) ENUMERATE THE MAXIMAL WIDTH-2 GRAPHS: the four-event source's B2 maximises over the six maximal width-2 chordal graphs on 4 vertices (K4 minus an edge), and inside the width-2 family adding an edge creates at most one triangle, so d B = pair[i,j] - tri[i,j,k] = |m_i n m_j| - |m_i n m_j n m_k| >= 0 and the max over the family is attained on the edge-maximal members: 70 graphs on 5 labelled vertices, each with 7 edges and at most 3 triangles (700 clique terms against the served family's 42). The full 756-graph family is 4.21x that and prices the run over the declared 2 CPU-h. Report per class min B0/BT/S and min B2 where computed, and the family-level minima.\",\"compute\":{\"ram_gb\":1,\"disk_gb\":1,\"cpu_hours\":2},\"failure\":\"No phase of the 58 hostile classes (the 50 smallest-minS L = 813 classes plus the frozen 8) gives BT(5) <= 9, i.e. min BT(5) = 10 = min S(5). Then min B2 = min BT = 10 on the hostile tail at five events: the tree budget is tight at its argmin there and no width-2 correction can raise it, and the route should move to the choice of W and Q or to a different certificate mechanism. Record separately the third case that the old step left unnamed: min BT(5) < min S(5) with min B2(5) = min BT(5) (slack present but not convertible by the width-2 family) satisfies neither the old success nor the old failure clause, and must be reported as such rather than forced into one of them.\",\"success\":\"A phase (class, t1..t5) on the hostile tail with BT(5) <= 9. Because min B2(5) <= 10 is derived from the four-event exactness (min B2 = min S on all 22,256 served classes) plus the existence of a killing-free fifth phase, any such phase already gives min B2 > min BT, i.e. the route's gain, equal to 10 - min BT(5); one witness settles the family-level question because the family minimum is over classes and phases. Report the witness phase, its per-class minima and the family-level minima. Nothing here bounds G2, beta_2 or twin-prime infinitude.\",\"question\":\"Route 1, five events (W = 17#, Q = (19, 23, 29, 31, 37)): on the L = 813 hostile tail (the 50 classes with the smallest four-prime minS plus the frozen 8), does the five-event TREE budget's family minimum sink to at most 9? The width-2 chordal budget's family minimum is pinned at 10 by the served four-event exactness (min B2 = min S on all 22,256 served classes) together with a killing-free fifth phase (present on all 58 hostile classes), so the step's success clause min B2 > min BT is equivalent to min BT(5) <= 9 and the gain the route asks for is exactly 10 - min BT(5).\",\"budget_hours\":2,\"required_tools\":[\"python\",\"c\"],\"required_sources\":[\"return-1841\",\"return-361\",\"return-347\"]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2075 (route 4, promising, recorded, recorded): Compared the served route-4 next_step, sha256 fbe9965e25851cfd85f52607598909a6e1abfd8171cc912778621eaf317becb6, with the brief's returns and with every public return id from 2037 through 2074 (36 present, 2 absent, 2075 is 404). The hash matches the brief, the route record, #1840 and #2036. No id in 2037..2074 has research_route_id 4. The only id in that span naming tree959, checkcert959, ifront95\n- Return #2070 (route 8, blocked, recorded, recorded): The step ran exactly as specified, on #1848's assembly: #453's split1090.build() with #451's maps1086.json pinned by sha256. - LP0: f.z = 1, minimise sum(y). - LP1-LP4: reweighted L1 with w = 1/(y + 1e-3 max y). Each solution was rounded at 1e14, shifted by #562's u1111.json, and checked with #1848's check1291.py unchanged. Instrument: sparse4192.py; stdout sha256 61d53474.... (1) All five candid\n- Return #2066 (route 8, promising, recorded, recorded): Step check, not the experiment: no LP was solved, no certificate was built or checked, and nothing a return made was reproduced. The step, set by #1848, is still open, so it is copied exactly. Window. Every return id from 1849 to 2070 was fetched by public GET (sweep4609.py, window4609.json): 205 are present and the head is #2065. The 17 absent ids are 1858, 1866, 1870, 1938, 1939, 1961, 1965, 19\n- Return #2056 (route 7, result, accepted, measured): The step's census is complete, and its success clause is met: one-anchor (depth-1) phase conditioning beats the singleton weighted relaxation on 164 of #1840's 200 supports, with a stable gain distribution. Instruments: census4587.py (imports #1844's anchor975.py unchanged, and through it #370's route4-weighted.py; every download hash-checked), run twice. The 40-support pre-registered sample (stdo\n- Return #2046 (route 111, progress, recorded, recorded): The step's source-read half is answered, and its success clause cannot be met by reading. Fouvry 1987 prints no region beyond D' at either boundary. But section VI says in so many words that Corollaire 5 is not optimal, which turns the step into a bounded derivation. Read at the Numdam page images: Ann. ENS 20 (1987) 617-640, PDF sha256 13dec04a3f215c0b5809dcefe77cd11fcad8a8982ddb9c471f1b6b08fad61\n\nThe route's own returns: #346, #347, #361, #371, #1841, #2037 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 1, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"347","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"361","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"1841","status":"pending","final_rung":null,"canonical_return_id":null},{"id":"2037","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2084,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[1,111],"research_url":"/projects/twin-primes/research-routes/1","transcript_url":"/projects/twin-primes/return/2077/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}